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Properties

Label 2-504-56.37-c1-0-25
Degree $2$
Conductor $504$
Sign $-0.754 + 0.656i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.267 − 1.38i)2-s + (−1.85 + 0.742i)4-s + (−1.56 − 0.902i)5-s + (2.63 − 0.217i)7-s + (1.52 + 2.38i)8-s + (−0.835 + 2.41i)10-s + (4.48 − 2.58i)11-s − 0.840i·13-s + (−1.00 − 3.60i)14-s + (2.89 − 2.75i)16-s + (−2.45 − 4.25i)17-s + (−4.87 − 2.81i)19-s + (3.57 + 0.515i)20-s + (−4.78 − 5.53i)22-s + (−3.05 + 5.28i)23-s + ⋯
L(s)  = 1  + (−0.188 − 0.981i)2-s + (−0.928 + 0.371i)4-s + (−0.698 − 0.403i)5-s + (0.996 − 0.0820i)7-s + (0.539 + 0.841i)8-s + (−0.264 + 0.762i)10-s + (1.35 − 0.779i)11-s − 0.232i·13-s + (−0.268 − 0.963i)14-s + (0.724 − 0.689i)16-s + (−0.595 − 1.03i)17-s + (−1.11 − 0.646i)19-s + (0.798 + 0.115i)20-s + (−1.02 − 1.17i)22-s + (−0.636 + 1.10i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.754 + 0.656i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.754 + 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368604 - 0.985492i\)
\(L(\frac12)\) \(\approx\) \(0.368604 - 0.985492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.267 + 1.38i)T \)
3 \( 1 \)
7 \( 1 + (-2.63 + 0.217i)T \)
good5 \( 1 + (1.56 + 0.902i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.48 + 2.58i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.840iT - 13T^{2} \)
17 \( 1 + (2.45 + 4.25i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.87 + 2.81i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.05 - 5.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.439iT - 29T^{2} \)
31 \( 1 + (3.66 + 6.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.56 - 2.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.23T + 41T^{2} \)
43 \( 1 + 7.34iT - 43T^{2} \)
47 \( 1 + (-2.83 + 4.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.15 + 0.669i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.31 + 4.22i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.77 - 2.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.647 + 0.373i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.08T + 71T^{2} \)
73 \( 1 + (-3.70 - 6.42i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.68 - 15.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.45iT - 83T^{2} \)
89 \( 1 + (-3.10 + 5.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.81T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.91844194105872896414764217188, −9.648234688111441517262785807078, −8.805071976003314026126473074941, −8.229849423578973051062795727732, −7.19318943257078116841251659137, −5.62933317523663261450202919723, −4.39974692012771943479219880135, −3.85049152109053134272997988436, −2.22727457060607016681911053471, −0.72891299584721825290914517388, 1.71374490544337510846081285203, 4.08146940013853675202825650375, 4.39520743206725685268439606440, 5.95688581538666247998314275380, 6.74821636908788362604007248197, 7.65033124881892961039046082447, 8.445063584786533358668868135591, 9.164861195971417259145014485526, 10.36878300174389416738769555787, 11.12919586865619129529768026396

Graph of the $Z$-function along the critical line